1. **Problem Statement:** A two-product firm faces the demand functions: $$q_1 = 40 - 2p_1 - p_2$$ $$q_2 = 35 - p_1 - p_2$$ and the cost function: $$C = 2q_1^2 + 2q_2^2 + 10$$ Find: (i) Output levels that satisfy the first order condition for maximum profit. (ii) Check the second order condition. (iii) Calculate the maximum profit. 2. **Step 1: Express prices in terms of quantities** From demand functions: $$q_1 = 40 - 2p_1 - p_2 \implies 2p_1 + p_2 = 40 - q_1$$ $$q_2 = 35 - p_1 - p_2 \implies p_1 + p_2 = 35 - q_2$$ Multiply second equation by 2: $$2p_1 + 2p_2 = 70 - 2q_2$$ Subtract first equation from this: $$(2p_1 + 2p_2) - (2p_1 + p_2) = (70 - 2q_2) - (40 - q_1)$$ $$p_2 = 30 - 2q_2 - 40 + q_1 = -10 - 2q_2 + q_1$$ Use $p_1 + p_2 = 35 - q_2$: $$p_1 = 35 - q_2 - p_2 = 35 - q_2 - (-10 - 2q_2 + q_1) = 35 - q_2 + 10 + 2q_2 - q_1 = 45 + q_2 - q_1$$ 3. **Step 2: Write profit function** Profit $\pi = p_1 q_1 + p_2 q_2 - C$ Substitute $p_1$ and $p_2$: $$\pi = (45 + q_2 - q_1) q_1 + (-10 - 2q_2 + q_1) q_2 - (2q_1^2 + 2q_2^2 + 10)$$ Expand: $$\pi = 45 q_1 + q_1 q_2 - q_1^2 - 10 q_2 - 2 q_2^2 + q_1 q_2 - 2 q_1^2 - 2 q_2^2 - 10$$ Combine like terms: $$\pi = 45 q_1 - 3 q_1^2 + 2 q_1 q_2 - 10 q_2 - 4 q_2^2 - 10$$ 4. **Step 3: First order conditions (FOC)** Set partial derivatives to zero: $$\frac{\partial \pi}{\partial q_1} = 45 - 6 q_1 + 2 q_2 = 0$$ $$\frac{\partial \pi}{\partial q_2} = 2 q_1 - 10 - 8 q_2 = 0$$ 5. **Step 4: Solve the system** From second equation: $$2 q_1 - 10 - 8 q_2 = 0 \implies 2 q_1 = 10 + 8 q_2 \implies q_1 = 5 + 4 q_2$$ Substitute into first equation: $$45 - 6(5 + 4 q_2) + 2 q_2 = 0$$ $$45 - 30 - 24 q_2 + 2 q_2 = 0$$ $$15 - 22 q_2 = 0 \implies 22 q_2 = 15 \implies q_2 = \frac{15}{22}$$ Then: $$q_1 = 5 + 4 \times \frac{15}{22} = 5 + \frac{60}{22} = 5 + \frac{30}{11} = \frac{55}{11} + \frac{30}{11} = \frac{85}{11}$$ 6. **Step 5: Check second order conditions (SOC)** Hessian matrix of $\pi$: $$H = \begin{bmatrix} \frac{\partial^2 \pi}{\partial q_1^2} & \frac{\partial^2 \pi}{\partial q_1 \partial q_2} \\ \frac{\partial^2 \pi}{\partial q_2 \partial q_1} & \frac{\partial^2 \pi}{\partial q_2^2} \end{bmatrix} = \begin{bmatrix} -6 & 2 \\ 2 & -8 \end{bmatrix}$$ Check definiteness: - Leading principal minor: $-6 < 0$ - Determinant: $(-6)(-8) - (2)(2) = 48 - 4 = 44 > 0$ Since first principal minor is negative and determinant positive, Hessian is negative definite, confirming a maximum. 7. **Step 6: Calculate maximum profit** Substitute $q_1 = \frac{85}{11}$ and $q_2 = \frac{15}{22}$ into profit: $$\pi = 45 q_1 - 3 q_1^2 + 2 q_1 q_2 - 10 q_2 - 4 q_2^2 - 10$$ Calculate each term: $$45 q_1 = 45 \times \frac{85}{11} = \frac{3825}{11}$$ $$-3 q_1^2 = -3 \times \left(\frac{85}{11}\right)^2 = -3 \times \frac{7225}{121} = -\frac{21675}{121}$$ $$2 q_1 q_2 = 2 \times \frac{85}{11} \times \frac{15}{22} = 2 \times \frac{1275}{242} = \frac{2550}{242} = \frac{1275}{121}$$ $$-10 q_2 = -10 \times \frac{15}{22} = -\frac{150}{22} = -\frac{75}{11}$$ $$-4 q_2^2 = -4 \times \left(\frac{15}{22}\right)^2 = -4 \times \frac{225}{484} = -\frac{900}{484} = -\frac{225}{121}$$ Sum all terms: $$\pi = \frac{3825}{11} - \frac{21675}{121} + \frac{1275}{121} - \frac{75}{11} - \frac{225}{121} - 10$$ Convert fractions to common denominator 121: $$\frac{3825}{11} = \frac{3825 \times 11}{11 \times 11} = \frac{42075}{121}$$ $$-\frac{75}{11} = -\frac{75 \times 11}{11 \times 11} = -\frac{825}{121}$$ $$-10 = -\frac{10 \times 121}{121} = -\frac{1210}{121}$$ Now sum: $$\pi = \frac{42075}{121} - \frac{21675}{121} + \frac{1275}{121} - \frac{825}{121} - \frac{225}{121} - \frac{1210}{121}$$ $$= \frac{42075 - 21675 + 1275 - 825 - 225 - 1210}{121} = \frac{42075 - 21675 + 1275 - 825 - 225 - 1210}{121}$$ Calculate numerator: $$42075 - 21675 = 20400$$ $$20400 + 1275 = 21675$$ $$21675 - 825 = 20850$$ $$20850 - 225 = 20625$$ $$20625 - 1210 = 19415$$ So: $$\pi = \frac{19415}{121} \approx 160.46$$ **Final answers:** (i) Output levels: $$q_1 = \frac{85}{11} \approx 7.73, \quad q_2 = \frac{15}{22} \approx 0.68$$ (ii) Second order condition satisfied (Hessian negative definite). (iii) Maximum profit: approximately 160.46.